On some classes of 2-dimensional Hermitian manifolds (Q1107143)

Let (M,J,g) be a 2-dimensional Hermitian manifold with Kähler form \(\Omega\) and Lee form \(\omega =\partial \Omega \circ J\). According to the symmetries of \(\nabla \omega\), \textit{F. Tricerri} and \textit{I. Vaisman} [Math. Z. 192, 205-216 (1986; Zbl 0603.53033)] defined and studied five basic classes \(\tau_ 1,...,\tau_ 5\) of (M,J,g). A sixth interesting class is that of generalized Hopf surfaces (g.H.s.), having \(\nabla \omega =0\), studied by \textit{I. Vaisman} [Geom. Dedicata 13, 231-255 (1982; Zbl 0506.53032)]. G.H.s. are endowed with four canonically defined Riemannian totally geodesic foliations, one of which is also complex analytic. In the present paper the author determines the geometric properties of the corresponding four canonically defined and generally non-integrable distributions, for (M,J,g) belonging to any of the classes \(\tau_ 1,...,\tau_ 5\) and to some subclasses of them.